Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q1,…,Qr be quadratic forms with real coefficients. We prove that the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{(Q_1(x),\ldots ,Q_r(x))\,\vert\, x\in{\Bbb Z}^s\}$\end{document} is dense in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Bbb R}^r$\end{document}, provided that the system Q1(x) = 0,…,Qr(x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q1,…,Qr are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form.